Related papers: On the Automorphisms of Order 15 for a Binary Self…
In this work, we introduce the concept of distance between self-dual codes, which generalizes the concept of a neighbor for self-dual codes. Using the k-neighbors, we are able to construct extremal binary self-dual codes of length 68 with…
By the Assmus and Mattson theorem, the codewords of each nontrivial weight in an extremal doubly even self-dual code of length 24m form a self-orthogonal 5-design. In this paper, we study the codes constructed from self-orthogonal 5-designs…
In this short note, we report the classification of self-dual $\mathbb{Z}_k$-codes of length $n$ for $k \le 24$ and $n \le 9$.
There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by generators and relations as $E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle.$ We study a special construction of self-orthogonal…
In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a…
Self-orthogonal codes are a significant class of linear codes in coding theory and have attracted a lot of attention. In \cite{HLL2023Te,LH2023Se}, $p$-ary self-orthogonal codes were constructed by using $p$-ary weakly regular bent…
We study the automorphisms of binary stabilizer codes and states. We prove that they almost always form a solvable group, and thereby shed new light on the fact that there is no universal set of transversal gates. We also determine the…
We classify finite subgroups of automorphism groups of non-trivial Severi--Brauer varieties of dimension $q-1$, where $q \geqslant 3$ is a prime number, over an arbitrary field. We also construct families of examples, namely, for every…
In the last 60 years coding theory has been studied a lot over finite fields $\mathbb{F}_q$ or commutative rings $\mathcal{R}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring…
Milliet asks the following question: given two prime numbers $p\neq q$, is there a division algebra of characteristic $p$ which is of dp-rank $q^2$ and of dimension $q^2$ over its center? We answer in the affirmative. We also give an…
In this paper we construct binary self-dual codes using the \'etale cohomology of $\mathbb{Z}/2$ on the spectra of rings of $S$-integers of global fields. We will show that up to equivalence, all self-dual codes of length at least 4 arise…
Binary self-orthogonal codes and balanced incomplete block designs are two combinatorial configurations that have been much studied because of their wide areas of application. In this paper, we have shown the distribution of (16; 6;…
In this paper, we study a special kind of factorization of $x^n+1$ over $\mathbb{F}_q, $ with $q$ a prime power $\equiv 3~({\rm mod}~4)$ when $n=2p,$ with $p\equiv 3~({\rm mod}~4)$ and $p$ is a prime. Given such a $q$ infinitely many such…
Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes…
We study a correspondence between orientation reversing involutions on compact 3-manifolds with only isolated fixed points and binary, self-dual codes. We show in particular that every such code can be obtained from such an involution. We…
In this paper, we construct a special family of cyclic codes, known as quadratic residue codes of prime length \( p \equiv \pm 1 \pmod{44} ,\) \( p \equiv \pm 5 \pmod{44} ,\) \( p \equiv \pm 7 \pmod{44} ,\) \( p \equiv \pm 9 \pmod{44} \)…
We construct a ternary [49,25,7] code from the row span of a Jacobsthal matrix. It is equivalent to a Generalized Quadratic Residue (GQR) code in the sense of van Lint and MacWilliams (1978). These codes are the abelian generalizations of…
In this note, the existence of noninner automorphisms of order 2 for finite 2-groups of coclass 2 is proved. Combining our result with a recent one due to Y. Guerboussa and M. Reguiat (see arXiv:1301.0085), we prove that every finite…
It is well known that every finite subgroup of automorphism group of polynomial algebra of rank 2 over the field of zero characteristic is conjugated with a subgroup of linear automorphisms. We prove that it is not true for an arbitrary…
We study the notion of formal-duality over finite cyclic groups of prime power order as introduced by Cohn, Kumar, Reiher and Sch\"urmann. We will prove that for any cyclic group of odd prime power order, as well as for any cyclic group of…